Abstract

We present results of a new analysis of the rotational kinematics of Titan, as constrained by Cassini radar data, extending over the entire currently available set of flyby encounters. Our analysis provides a good constraint on the current orientation of the spin pole, but does not have sufficient accuracy and duration to clearly see the expected spin pole precession. In contrast, we do clearly see temporal variations in the spin rate, which are driven by gravitational torques which attempt to keep the prime meridian oriented toward Saturn.

Titan is a synchronous rotator. At lowest order, that means that the rotational and orbital motions are synchronized. At the level of accuracy required to fit the Cassini radar data, we can see that synchronous rotation and uniform rotation are not quite the same thing. Our best fibng model has a fixed pole, and a rotation rate which varies with time, so as to keep Titan’s prime meridian oriented towards Saturn, as the orbit varies.

A gravitational torque on the tri-axial figure of Titan attempts to keep the axis of least inertia oriented toward Saturn. The main effect is to synchronize the orbit and rotation periods, as seen in inertial space. The response of the rotation angle, to periodic changes in orbital mean longitude, is modeled as a damped, forced harmonic oscillator. This acts as a low-pass filter. The rotation angle accurately tracks orbital variations at periods longer than the free libration period, but is unable to follow higher frequency variations.

The mean longitude of Titan’s orbit varies on a wide range of time scales. The largest variations are at Saturn’s orbital period (29.46 years), and are due to solar torques. There are also variations at periods of 640 and 5800 days, due to resonant interaction with Hyperion.

For a rigid body, with moments of inertia estimated from observed gravity, the free libration period for Titan would be 850 days. The best fit to the radar data is obtained with a libration period of 645 days, and a damping time of 1000 years.

The principal deviation of Titan’s rotation from uniform angular rate, as seen in the Cassini radar data, is a periodic signal resonantly forced by Hyperion.

Notes

Titan:

hard to see surface

Cassini’s radar intended for mapping surface

didn’t get much by way of repeat observations (“tie points”), which are needed to constrain rotation

Session: Moon Formation and Dynamics I

Henry C. (Luke) Dones (SWRI)

Abstract

Charnoz et al. (2010) proposed that Saturn’s small “ring moons” out to Janus and Epimetheus consist of ring material that viscously spread beyond the Roche limit and coagulated into moonlets. The moonlets evolve outward due to the torques they exert at resonances in the rings. More massive moonlets migrate faster; orbits can cross and bodies can merge, resulting in a steep trend of mass vs. distance from the planet. Canup (2010) theorized that Saturn’s rings are primordial and originated when a differentiated, Titan-like moon migrated inward when the planet was still surrounded by a gas disk. The satellite’s icy shell could have been tidally stripped, and would have given rise to today’s rings and the mid-sized moons out to Tethys. Charnoz et al. (2011) investigated the formation of satellites out to Rhea from a spreading massive ring, and Crida and Charnoz (2012) extended this scenario to other planets. Once the mid-sized moons recede far from the rings, tidal interaction with the planet determines the rate at which the satellites migrate. Charnoz et al. (2011) found that Mimas would have formed about 1 billion years more recently than Rhea. The cratering records of these moons (Kirchoff and Schenk 2010; Robbins et al. 2015) provide a test of this scenario. If the mid-sized moons are primordial, most of their craters were created through hypervelocity impacts by ecliptic comets from the Kuiper Belt/Scattered Disk (Zahnle et al. 2003; Dones et al. 2009). In the Charnoz et al. scenario, the oldest craters on the moons would result from low-speed accretionary impacts. We thank the Cassini Data Analysis program for support.

Abstract

The variation of a planet’s obliquity is influenced by the existence of satellites with a high mass ratio. For instance, the Earth’s obliquity is stabilized by the Moon, and would undergo chaotic variations in the Moon’s absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically-forced climate change is the rate of chaotic diffusion. Accordingly, here we reexamine the spin-axis evolution of a Moonless Earth within the context of a simplified perturbative framework. We present analytical estimates of the characteristic Lyapunov coefficient as well as the chaotic diffusion rate and demonstrate that even in absence of the Moon, the stochastic change in the Earth’s obliquity is sufficiently slow to not preclude long-term habitability. Our calculations are consistent with published numerical experiments and illustrate the putative system’s underlying dynamical structure in a simple and intuitive manner. In addition, we examine if at any point in the Earth’s evolutionary history, the obliquity varied significantly. We find that even though the orbital perturbations were different in the past, the system nevertheless avoided resonant encounters throughout its evolution. This indicates that the Earth obtained its current obliquity during the formation of the Moon.

Session: Moon Formation and Dynamics III

Maja Cuk (SETI Institute)

Abstract

Mimas and Enceladus are the smallest and innermost mid-sized icy moons of Saturn. They are each caught in a 2:1 orbital resonance with an outer, larger moon: Mimas with Tethys, Enceladus with Dione. This is where the similarities end. Mimas is heavily cratered and appears geologically inactive, while Enceladus has a young surface and high tidal heat flow. Large free eccentricity of Mimas implies low tidal dissipation, while Enceladus appears very dissipative, likely due to an internal ocean. Their resonances are very different too. Mimas is caught in a 4:2 inclination type resonance with Tethys which involves inclinations of both moons. Enceladus is in a 2:1 resonance with Dione which affects only Enceladus’s eccentricity. The well-known controversy over the heat flow of Enceladus can be solved by invoking a faster tidal evolution rate than previously expected (Lainey et al. 2012), but other mysteries remain. It has been long known that Mimas has very low probability of being captured into the present resonance, assuming that the large resonant libration amplitude reflects sizable pre-capture inclination of Mimas. Furthermore, Enceladus seems to have avoided capture into a number of sub-resonances that should have preceded the present one. An order of magnitude increase in the rate of tidal evolution does not solve these problems. It may be the time to reconsider the dominance of tides in the establishment of these resonances, especially if the moons themselves may be relatively young. An even faster orbital evolution due to ring/disk torques can help avoid capture into smaller resonances. Additionally, past interaction of Mimas with Janus and Epimetheus produce some of the peculiarities of Mimas’ current orbit. At the meeting I will present numerical integrations that confirm the the existence of these problems, and demonstrate the proposed solutions.

Abstract

The formation of Pluto’s small satellites – Styx, Nix, Keberos and Hydra remains a mystery. Their orbits are nearly circular (eccentricity $e = 0.0055$ or less) and near resonances and coplanar with respect to Charon. One scenario suggests that they all formed close to their current locations from a disk of debris, which was ejected from the Charon-forming impact. We test the validity of this scenario by performing N-body simulations with Pluto-Charon evolving tidally from an initial orbit at a few Pluto radii. The small satellites are modeled as test particles with initial orbital distances within the range of the current small satellites and damped to their coldest orbits by collisional damping. It is found that if Charon is formed from a debris disk and has low initial eccentricity, all test particles survive to the end of the tidal evolution, but there is no preference for resonances and the test particles’ final $e$ is typically > 0.01. If Charon is formed in the preferred intact capture scenario and has initial orbital eccentricity ~ 0.2, the outcome depends on the relative rate of tidal dissipation in Charon and Pluto, $A$. If $A$ is large and Charon’s orbit circularizes quickly, a significant fraction of the test particles survives outside resonances with $e \gtrsim 0.01$. Others are ejected by resonance or survive in resonance with very large $e$ (> 0.1). If $A$ is small and Charon’s orbit remains eccentric throughout most of the tidal evolution, most of the test particles are ejected. The test particles that survive have $e \gtrsim 0.01$, including some with $e \gt 0.1$. None of the above cases results in test particles with sufficiently low final $e$.

Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

Notes

Libration measurement

3D reconstruction of coords of a network of control point (fiducial satellite surface points — e.g. craters)

Session: Moon Formation and Dynamics II

Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

Session: Moon Formation and Dynamics I

Valeri Makarov (for Bryan Dorland) (USNO)

Abstract

Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the actual orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. We contend that both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the tidal torque and the triaxial torque, the latter related to a permanent deformation of the distribution of mass from a perfect rotational symmetry. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of the semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture of thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.

Notes

Numerical sim of Lunarspinlibrations in longitude with polar torques exerted by Earth

spectrum of harmonics

dominant term: $\dfrac{\dot{\theta}}{n}\, – 1 \propto -\cos M$

Geometry offorcedlibrations

longest axis of planet tries to align with line of centers (but can’t)

Abstract

The Fermi Bubbles are large structures that stretch symmetrically between galactic latitudes of -55 degrees and +55 degrees and between galactic longitudes of -45 degrees and +45 degrees. For almost a decade they have been under the intense scrutiny of the Fermi-Large Area Telescope, a gamma-ray detector in orbit about the earth. The Bubbles remain mysterious: are the gamma-rays – with energies up to a few hundred GeV – produced by hadrons or do they come from inverse Compton scattering of galactic electrons with the low energy interstellar radiation field? Why are the edges of the bubbles only 3 degree wide? How old are the bubbles? For some time we have been considering a non-Newtonian cosinusoidal potential $U=-\dfrac{G M}{r} \cos(k_0 r)$, and its complement, a non-Coulombic electric potential $U=Q \exp(-k_0 r)$. In both cases, $k_0 = 2 \pi/400$ pc. In this talk we present evidence that our putative potentials acting in concert can help answer the mysteries of the Bubbles.

Abstract

p-ellipses are simple, yet very accurate formulae for orbits in power-law potentials, like those approximating galaxy disks. These precessing elliptical orbits reveal important systematics of orbits in such potentials, including simple expressions for the dependence of apsidal precession on eccentricity, and the fact that very few terms (or parameters) are needed for the approximation of even nearly radial orbits. The orbit approximations are also useful tools for addressing problems in galaxy dynamics. In particular, they indicate the existence of a range of eccentric resonances associated with the usual, near-circular Lindblad resonances. Collectively these change an isolated Lindblad resonance to a Lindblad Zone of eccentric resonances. A range of these resonances could be excited at a common paRern speed, aiding the formation of a variety of bars and spirals, out of eccentric orbits. Such waves would be persistent, and not wind up or disperse, since differences in their precession frequencies offset differences in the circular velocities at the radii of their parent orbits. The p-ellipse approximation further reveals how a non-axisymmetric component of the gravitational potential (e.g., due to bar self-gravity) significantly modifies precession frequencies, and similarly modifies the Lindblad Zones.

Fred C. Adams (U. Michigan)

Abstract

Numerous spectroscopic and photometric studies have provided strong evidence of the presence of multiple stellar populations in globular clusters and raised many fundamental questions concerning the formation and dynamical evolution of these stellar systems. After a brief review of the main observational studies, I will present the results of theoretical investigations exploring a number of aspects of the internal dynamics of multiple-population clusters and their formation history. Most planetary systems are formed within stellar clusters, and these environments can shape their properties. This talk considers scattering encounters between solar systems and passing cluster members, and calculates the corresponding interaction cross sections. The target solar systems are generally assumed to have four giant planets, with a variety of starting states, including circular orbits with the semimajor axes of our planets, a more compact configuration, an ultracompact state with multiple mean motion resonances, and systems with massive planets. We then consider the effects of varying the cluster velocity dispersion, the relative importance of binaries versus single stars, different stellar host masses, and finite starting eccentricities of the planetary orbits. For each state of the initial system, we perform an ensemble of numerical scaRering experiments and determine the cross sections for eccentricity increase, inclination angle increase, planet ejection, and capture. This talk reports results from over 2 million individual scattering simulations. Using supporting analytic considerations, and fibng functions to the numerical results, we find a universal formula that gives the cross sections as a function of stellar host mass, cluster velocity dispersion, starting planetary orbital radius, and final eccentricity. The resulting cross sections can be used in a wide variety of applications. As one example, we revisit constraints on the birth aggregate of our Solar System due to dynamical scattering and find N < 10,000 (consistent with previous estimates).

Session: Star Cluster and Galaxy Dynamics

Enrico Vesperini (Indiana University) (invited) [withdrawn]

Abstract

Numerous spectroscopic and photometric studies have provided strong evidence of the presence of multiple stellar populations in globular clusters and raised many fundamental questions concerning the formation and dynamical evolution of these stellar systems. After a brief review of the main observational studies, I will present the results of theoretical investigations exploring a number of aspects of the internal dynamics of multiple-population clusters and their formation history.

Session: Dynamics of Small Solar System Bodies III

Victor J. Slabinski (USNO)

Abstract

The Lense‑Thirring (L‑T) effect from General Relativity predicts a small secular increase to the node right ascension for close Earth satellites. For the LAGEOS 1 satellite, the predicted node increase is 31 mas/y. There is a current effort to observationally evaluate L‑T to 1 percent accuracy through an orbit analysis of the laser‑ranged LAGEOS 1, LAGEOS 2, and LARES satellites. Uncertainty in the computed gravitational perturbations to the satellite nodes, due to parameter uncertainties, is largely eliminated by taking a linear combination of the node positions which eliminates the uncertainty due to the major terms. One then looks for the L‑T effect on this composite node.

But there remains uncertainty in the computed perturbations due to two radiation (non‑gravitational) forces: the solar radiation (SR) force and thermal thrust (Yarkovsky effects). This paper treats LAGEOS 1 perturbations. For simplicity in discussion, we treat perturbations to its node rather than perturbations to the composite node.

Uncertainty in the perturbation rates arises from ignorance of parameter values for the LAGEOS 1 exterior aluminum surface, specifically, the solar absorbtance and thermal emiRance. The LAGEOS 1 Phase B design study proposed three different sets of aluminum surface parameters without recommending a particular set. The LAGEOS 1 as-built surface parameters were not measured prior to spacecraft launch.

The possible spread in LAGEOS 1 solar absorbtance values gives a spread of ±0.42 mas/y in the SR force contribution to its node rate. This results in a ±1.3 percent uncertainty to the L‑T determination. But because of its long‑period perturbation to the eccentricity vector, evaluating the SR force parameter as a solved‑for parameter in the orbit analysis should significantly reduce the uncertainty in the corresponding node motion. The possible spread in LAGEOS 1 surface values gives a spread of ±0.16 mas/y in the thermal thrust contribution to its node rate. This represents a ±0.53 percent uncertainty in the L‑T determination which leaves little room for other error sources. Ground-based satellite brightness measurements could improve knowledge of the surface absorbtance and reduce the uncertainty from thermal thrust.

Notes

Lense-Thirring

gravitomagnetic effect

spinning Earth:

$\rightarrow$ frame-dragging

$\rightarrow$ precession of $\Omega$ and $\omega$

LAGEOS 1 & 2: linear motion of $\Omega \approx 1.8$ m/yr

Goal: 1% measurement of L-T effect

Other perturbing forces

Solar radiation pressure

requires knowledge of satellite surface material properties

notably: aging

Thermal thrust

IR from Earth

fused silica of corner-cube reflectors is an excellent absorber of IR

Oops

thermal phase lag: max recoil force not at local midnight but somewhat past